| L(s) = 1 | − 5·7-s − 2·11-s + 2·13-s − 2·17-s − 5·19-s + 6·23-s + 5·25-s + 16·29-s + 3·31-s + 9·37-s − 4·41-s + 2·43-s + 8·47-s + 18·49-s + 6·53-s + 6·59-s + 2·61-s + 5·67-s − 8·71-s + 11·73-s + 10·77-s + 5·79-s + 12·89-s − 10·91-s + 36·97-s − 6·101-s + 11·103-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 0.603·11-s + 0.554·13-s − 0.485·17-s − 1.14·19-s + 1.25·23-s + 25-s + 2.97·29-s + 0.538·31-s + 1.47·37-s − 0.624·41-s + 0.304·43-s + 1.16·47-s + 18/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s + 0.610·67-s − 0.949·71-s + 1.28·73-s + 1.13·77-s + 0.562·79-s + 1.27·89-s − 1.04·91-s + 3.65·97-s − 0.597·101-s + 1.08·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106699033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106699033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.210220126892870990936180394985, −8.802887512635859271236806134655, −8.729983498780691958254515901069, −8.428174046964265616624622357564, −7.57565832573131320214356719704, −7.51874296049311745459624114336, −6.67866005004160149304255744705, −6.63368033627424095526584596572, −6.29429945650867780502041776543, −6.04700827487207487213898640207, −5.20647901203443763894440876646, −5.01781905411925733435518817575, −4.30573812498332139594262724586, −4.14319712929519341650642299714, −3.23645231162302387639856389423, −3.18319680120722497899312602920, −2.39035303845600298420578278056, −2.38941155371295873847834442547, −0.835380808261731764761490791627, −0.75548003203376718681735610263,
0.75548003203376718681735610263, 0.835380808261731764761490791627, 2.38941155371295873847834442547, 2.39035303845600298420578278056, 3.18319680120722497899312602920, 3.23645231162302387639856389423, 4.14319712929519341650642299714, 4.30573812498332139594262724586, 5.01781905411925733435518817575, 5.20647901203443763894440876646, 6.04700827487207487213898640207, 6.29429945650867780502041776543, 6.63368033627424095526584596572, 6.67866005004160149304255744705, 7.51874296049311745459624114336, 7.57565832573131320214356719704, 8.428174046964265616624622357564, 8.729983498780691958254515901069, 8.802887512635859271236806134655, 9.210220126892870990936180394985
